Non Linear Operators

Abstract

Linear operators are the simplest operators. In many problems one has to consider more complicated nonlinear operators. As in the case of linear operators, again the main problem is to solve equations Ax = y for a nonlinear A in a Hilbert or Banach space. Geometrically, this problem means that a certain map or operator B leaves fixed at least one vector x, i.e., \[x = Bx, where Bx = x + Ax - y\], and we have to find this vector. Theorems which establish the existence of such fixed vectors are called fixed point theorems. There are a number of very important fixed point theorems. In this chapter we present one of the simplest; the Contraction Mapping Theorem. This theorem is very powerful in that it allows one to prove the existence of solutions to nonlinear integral, differential and functional equations, and it gives a procedure for numerical approximations to the solution. Some of the applications are also included in this chapter.